{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:UPGYGCZWZ4FTD7ZWPKNTM6I2LZ","short_pith_number":"pith:UPGYGCZW","canonical_record":{"source":{"id":"1208.5041","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-08-24T19:39:20Z","cross_cats_sorted":["math.CO","math.MG"],"title_canon_sha256":"35c2e5862bd66b182de25443aa39c33c5c8604dd11d461a1fdf524ea60fff1cc","abstract_canon_sha256":"76b7f6a61bf272e2650882da48c5c81a2964a0c920c7c37b595bcfbff38778f4"},"schema_version":"1.0"},"canonical_sha256":"a3cd830b36cf0b31ff367a9b36791a5e7104d40001d5a466b4c6ea7346cb0ae7","source":{"kind":"arxiv","id":"1208.5041","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.5041","created_at":"2026-05-18T03:47:52Z"},{"alias_kind":"arxiv_version","alias_value":"1208.5041v1","created_at":"2026-05-18T03:47:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.5041","created_at":"2026-05-18T03:47:52Z"},{"alias_kind":"pith_short_12","alias_value":"UPGYGCZWZ4FT","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_16","alias_value":"UPGYGCZWZ4FTD7ZW","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_8","alias_value":"UPGYGCZW","created_at":"2026-05-18T12:27:23Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:UPGYGCZWZ4FTD7ZWPKNTM6I2LZ","target":"record","payload":{"canonical_record":{"source":{"id":"1208.5041","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-08-24T19:39:20Z","cross_cats_sorted":["math.CO","math.MG"],"title_canon_sha256":"35c2e5862bd66b182de25443aa39c33c5c8604dd11d461a1fdf524ea60fff1cc","abstract_canon_sha256":"76b7f6a61bf272e2650882da48c5c81a2964a0c920c7c37b595bcfbff38778f4"},"schema_version":"1.0"},"canonical_sha256":"a3cd830b36cf0b31ff367a9b36791a5e7104d40001d5a466b4c6ea7346cb0ae7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:47:52.035181Z","signature_b64":"FG/Fp0z/bTjGePWii88Y7H3V8QZLNhBR+EZW9vm/aEo0mxdyOUMrJ1wjFIHT/0laHLs0XonGg9+yC0at6thTCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a3cd830b36cf0b31ff367a9b36791a5e7104d40001d5a466b4c6ea7346cb0ae7","last_reissued_at":"2026-05-18T03:47:52.034706Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:47:52.034706Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1208.5041","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:47:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ijkk/SpVJ4K91EkFhxPIG4MBV+wMlVoc6sTaFl8Js4lIiQIfKae37sM8YQe0WzRtk8Y4Kuhd7DbIsBW8ES0ZAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T06:25:12.274072Z"},"content_sha256":"bfca495126d25aa7529fa96bcc88de46c98bd26a252b12074073fedec963af23","schema_version":"1.0","event_id":"sha256:bfca495126d25aa7529fa96bcc88de46c98bd26a252b12074073fedec963af23"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:UPGYGCZWZ4FTD7ZWPKNTM6I2LZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On subsets of $S^n$ whose $(n+1)$-point subsets are contained in open hemisheres","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.MG"],"primary_cat":"math.GR","authors_text":"Brendan Owens, Peter Kropholler, Robert Bieri","submitted_at":"2012-08-24T19:39:20Z","abstract_excerpt":"We investigate the nature of subsets of spheres which satisfy a tameness condition associated with the Bieri-Groves conjecture on cohomological finiteness conditions for metabelian groups. We find that there is a natural polyhedrality in a crucial special case. In the case of the two dimensional sphere we establish a strong polyhedrality condition for certain open sets which are maximal subject to satisfying the tameness condition that subsets of three or fewer points are contained in open hemispheres. Many examples are included."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.5041","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:47:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vs0apqnnfbBSKuObyUbMLSzENDhkXE2Zcv3hWK1rsGgRNdj5BI57AZuG2j5Nq1Swq7p0+Cy7Y3vHPzVv7mxXDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T06:25:12.274432Z"},"content_sha256":"7f024c5cac8902d9ae3d6d35a3774f7e1a684c7fe6af5e5eab3254c202f9fcc0","schema_version":"1.0","event_id":"sha256:7f024c5cac8902d9ae3d6d35a3774f7e1a684c7fe6af5e5eab3254c202f9fcc0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UPGYGCZWZ4FTD7ZWPKNTM6I2LZ/bundle.json","state_url":"https://pith.science/pith/UPGYGCZWZ4FTD7ZWPKNTM6I2LZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UPGYGCZWZ4FTD7ZWPKNTM6I2LZ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-28T06:25:12Z","links":{"resolver":"https://pith.science/pith/UPGYGCZWZ4FTD7ZWPKNTM6I2LZ","bundle":"https://pith.science/pith/UPGYGCZWZ4FTD7ZWPKNTM6I2LZ/bundle.json","state":"https://pith.science/pith/UPGYGCZWZ4FTD7ZWPKNTM6I2LZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UPGYGCZWZ4FTD7ZWPKNTM6I2LZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:UPGYGCZWZ4FTD7ZWPKNTM6I2LZ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"76b7f6a61bf272e2650882da48c5c81a2964a0c920c7c37b595bcfbff38778f4","cross_cats_sorted":["math.CO","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-08-24T19:39:20Z","title_canon_sha256":"35c2e5862bd66b182de25443aa39c33c5c8604dd11d461a1fdf524ea60fff1cc"},"schema_version":"1.0","source":{"id":"1208.5041","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.5041","created_at":"2026-05-18T03:47:52Z"},{"alias_kind":"arxiv_version","alias_value":"1208.5041v1","created_at":"2026-05-18T03:47:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.5041","created_at":"2026-05-18T03:47:52Z"},{"alias_kind":"pith_short_12","alias_value":"UPGYGCZWZ4FT","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_16","alias_value":"UPGYGCZWZ4FTD7ZW","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_8","alias_value":"UPGYGCZW","created_at":"2026-05-18T12:27:23Z"}],"graph_snapshots":[{"event_id":"sha256:7f024c5cac8902d9ae3d6d35a3774f7e1a684c7fe6af5e5eab3254c202f9fcc0","target":"graph","created_at":"2026-05-18T03:47:52Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We investigate the nature of subsets of spheres which satisfy a tameness condition associated with the Bieri-Groves conjecture on cohomological finiteness conditions for metabelian groups. We find that there is a natural polyhedrality in a crucial special case. In the case of the two dimensional sphere we establish a strong polyhedrality condition for certain open sets which are maximal subject to satisfying the tameness condition that subsets of three or fewer points are contained in open hemispheres. Many examples are included.","authors_text":"Brendan Owens, Peter Kropholler, Robert Bieri","cross_cats":["math.CO","math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-08-24T19:39:20Z","title":"On subsets of $S^n$ whose $(n+1)$-point subsets are contained in open hemisheres"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.5041","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:bfca495126d25aa7529fa96bcc88de46c98bd26a252b12074073fedec963af23","target":"record","created_at":"2026-05-18T03:47:52Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"76b7f6a61bf272e2650882da48c5c81a2964a0c920c7c37b595bcfbff38778f4","cross_cats_sorted":["math.CO","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-08-24T19:39:20Z","title_canon_sha256":"35c2e5862bd66b182de25443aa39c33c5c8604dd11d461a1fdf524ea60fff1cc"},"schema_version":"1.0","source":{"id":"1208.5041","kind":"arxiv","version":1}},"canonical_sha256":"a3cd830b36cf0b31ff367a9b36791a5e7104d40001d5a466b4c6ea7346cb0ae7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a3cd830b36cf0b31ff367a9b36791a5e7104d40001d5a466b4c6ea7346cb0ae7","first_computed_at":"2026-05-18T03:47:52.034706Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:47:52.034706Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FG/Fp0z/bTjGePWii88Y7H3V8QZLNhBR+EZW9vm/aEo0mxdyOUMrJ1wjFIHT/0laHLs0XonGg9+yC0at6thTCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:47:52.035181Z","signed_message":"canonical_sha256_bytes"},"source_id":"1208.5041","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bfca495126d25aa7529fa96bcc88de46c98bd26a252b12074073fedec963af23","sha256:7f024c5cac8902d9ae3d6d35a3774f7e1a684c7fe6af5e5eab3254c202f9fcc0"],"state_sha256":"b2a06845b031d5278da0e6a3214941bda088f2e0bdc03f502179f7b6cbe61d25"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sHSATSGMk/OJiXfYP5/cv/yHdiHT3tLHONiU6KOqxeYszoz9V1Sz6Fduw6xcpmHHx/3qR+9QX5g5ESFKZqWHBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T06:25:12.276375Z","bundle_sha256":"b45a018bf4c4f8a66bb1218fb5ded99117b79198a14a86847875d11750e9fdf5"}}